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Question 10289: Validity of a Finite Total-Function Graph

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0019-10289 · q-20261003-r06-function-graph-0089

Original question and givens

The declared domain is D={9, 10} and codomain is B={3, 4, 5}. Is the ordered-pair set {(9, 3), (10, 43)} exactly the graph of a total function D→B?

  • G001 — The declared domain is D={9, 10} and codomain is B={3, 4, 5}. Is the ordered-pair set {(9, 3), (10, 43)} exactly the graph of a total function D→B?

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

Step-by-step derivation

S001 · implies

Before

raw domain=[9, 10]

After

sorted distinct declared domain=[9, 10]

Conditions: none

The domain is the declared finite integer subset,not automatically all R.The input list must already be distinct.The first coordinate is input,the second output.

Step identifiers and dependencies

Rule: function_graph_declared_domain_scope · Uses: G001

S002 · implies

Before

codomain=[3, 4, 5];raw pair rows=[[9, 3], [10, 43]]

After

ordered-pair set=[[9, 3], [10, 43]]

Conditions: none

This protocol treats the graph as a set:deduplicate identical pairs,but retain different y values at one x.This differs from a raw-row protocol requiring one original row per input.

Step identifiers and dependencies

Rule: graph_set_duplicate_pairs_count_once · Uses: G001, S001

S003 · implies

Before

pair=[9, 3];domain=[9, 10];codomain=[3, 4, 5]

After

both coordinates typed=Yes

Conditions: none

Every pair must lie in D×B.Do not ignore outside-domain or outside-codomain pairs after finding legal points for some inputs.

Step identifiers and dependencies

Rule: graph_every_pair_type_guard · Uses: G001, S002

S004 · implies

Before

pair=[10, 43];domain=[9, 10];codomain=[3, 4, 5]

After

both coordinates typed=No

Conditions: none

Every pair must lie in D×B.Do not ignore outside-domain or outside-codomain pairs after finding legal points for some inputs.

Step identifiers and dependencies

Rule: graph_every_pair_type_guard · Uses: G001, S003

S005 · implies

Before

declared input=9;normalized pair set=[[9, 3], [10, 43]]

After

distinct output fiber=[3];count=1

Conditions: none

Each declared input needs exactly one distinct output.At most one point gives single-valuedness,not totality.

Step identifiers and dependencies

Rule: graph_each_declared_input_exactly_one · Uses: G001, S004

S006 · implies

Before

declared input=10;normalized pair set=[[9, 3], [10, 43]]

After

distinct output fiber=[43];count=1

Conditions: none

Each declared input needs exactly one distinct output.At most one point gives single-valuedness,not totality.

Step identifiers and dependencies

Rule: graph_each_declared_input_exactly_one · Uses: G001, S005

S007 · implies

Before

all pair type flags=[Yes, No];source fiber sizes=[1, 1]

After

total function graph=No

Conditions: none

Typing and totality are joint conditions.An empty domain with an empty graph is valid,but extra outside pairs are not.The valid empty function does not require a nonempty codomain.

Step identifiers and dependencies

Rule: total_graph_type_and_totality_joint_gate · Uses: G001, S006

Reverse checks

  • S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

No

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: 61fd7ba81f09c323931cb15fc7e1e0899bfd48c6c159aeb4137802c8b68d2e58
  • candidate record SHA-256: 04947d164d8bdceb7c0e7fb20325bcb437862f8e5bda7708d18db5c33f0d4077
  • semantic review receipt SHA-256: eeb35dc5daab92234bf72827627392b47f38ead4cfb574d4373bebc43c09fbf5
  • question-first witness SHA-256: 3cf8e0d3a00b0df1994db2bfde81f4a31b382e170bdda83ea3d0b68da64e8419
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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